This invention relates generally to apparatus and methods for analyzing and comparing signals, generally of the continuous waveform type, for the purpose of deriving information from the differences between the signals. Such analysis and comparison may be made, for example, with respect to signals which have been transmitted from a signal source through a medium for deriving information concerning the medium from the detected differences between the transmitted and received signals. Such differences in transmitted and received signals may occur, for example, as a result of certain characteristics of the medium which may include inhomogeneities and/or non-uniformities in the medium. More particularly, the present invention relates to the analysis and comparison of signals utilizing entropy based analytical apparatus and methods.
Types of apparatus and methods of the foregoing type include radar and sonar systems wherein continuous waveform signals are transmitted through media such as the earth's atmosphere or water and are used to probe these media for discontinuities representing objects, inhomogeneities or disturbances of various types. The use of probing signals for the identification of objects, inhomogeneities, discontinuities, non-uniformities or disturbances in wave propagation media has been known in the art for a wide variety of applications.
In prior art systems of this type, such as in early radar and sonar systems, it was common to utilize simple but effective square-law detection and envelope detection techniques. In later systems, so-called correlation processor systems, which computed the correlation of received signals with replicas of the transmitted signal, were utilized and were considered more effective for some applications in extracting the returning signals from background noise and other forms of interference.
The square-law detector, the envelope detector and the correlation detector all share a common attribute in that they all calculate a quantity proportional to the energy of the received signal. In the case of received signals that are replicas of the transmitted signal which are obtained, for example, by reflection from a plane surface, the measurement of signal energy as the means of detection yields, according to well supported theory, the highest possible signal to noise ratios possible for a linear time-shift-invariant signal processing algorithm. However, recent experimental results have indicated that entropy measurements are significantly more effective than energy measurements as a means for detecting and analyzing the characteristics of the received signals.
In this respect, the utility of entropy imaging has been disclosed, for example, in the following scientific papers: (1) "Analysis of digitized waveforms using Shannon entropy" by M. S. Hughes, JASA 93(2), Feb. 1993, pp 892-906, (2) "A comparison of Shannon entropy versus signal energy for acoustic detection of artificially induced defects in Plexiglass" by M. S. Hughes, JASA 91(4) Pt.1 April 1992, pp 2272-2275 and (3) "Analysis of Ultrasonic waveforms using Shannon entropy", by M. S. Hughes, Proceedings IEEE UFFC symposium 1992, pp 1205-1209, all of which were authored by the present inventor and which are incorporated herein by reference. Apparatus and methods for such entropy based signal analysis techniques are also disclosed in U.S. patent application Ser. No. 07/906,571 entitled ENTROPY-BASED SIGNAL RECEIVER filed Jun. 30, 1992, in the name of the inventor of the subject matter of the present application, Michael S. Hughes.
In such previously reported entropy based systems and methods, it is necessary to calculate the "density distribution function" w(y) of the signal being analyzed in order to determine its entropy. A rigorous definition of w(y) is given in the paper of reference (1) above and is also explained further in the other references noted. The density distribution function w(y) is, in very general terms, a measure of how often in a selected time period chosen for purposes of analysis the function being analyzed takes on a selected value Y.sub.1. For purposes of making such calculations, the signal waveform being analyzed is first digitized over the selected time interval in which the analysis is to be made. Then, as described in the references given above, a Fourier series approach is used to calculate the function w(y).
In such Fourier series based methods, it was found necessary to compute the density distribution function w(y) over an interval greater than the actual range of the received time dependent signal f(t), that is, over a range greater than [f.sub.min,f.sub.max ] of f(t). This was found necessary in order to prevent edge effects from corrupting the estimate of w(y) over the actual range [f.sub.min,f.sub.max ] of f(t). This imposes additional computational burdens on the apparatus and methods based on the Fourier series approach.
Secondly, the size of the computation required in the Fourier series approach is governed primarily by (1) the number of terms N.sub.co in the Fourier series required to represent w(y) and (2) the number of digitized points N.sub..alpha. used to represent the underlying continuous waveform. More precisely, the Fourier series approach requires the evaluation of N.sub.co .times.N.sub..alpha. double precision sums. This requires a substantial computational size for most applications.
A typical Fourier series method as previously reported is illustrated in steps 146 through 156 in the flow diagram FIG. 8. The details of the steps presented in FIG. 8 are disclosed in the previously published work of the present inventor and no further explanation of these steps is therefore required.
Thus, the Fourier series entropy based approach, while yielding acceptable results for most applications, requires a substantial computational size for its implementation. Certain other aspects of the Fourier series approach will also be further analyzed below.